Subgroups — #21436817

Order 7552.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838124	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838067	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837764	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837763	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837681	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837620	C₂	1	1	yes	maximal, characteristic
9	#16837618	C₂	1	1	yes	maximal, characteristic
10	#16837606	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837500	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837484	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	—	2	1	no	—
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745671	C₂²	1	1	yes	characteristic
19	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273725	C₂²	1	1	yes	characteristic
23	#14273722	—	2	1	no	—
24	#14273721	—	2	1	no	—
25	#14273716	C₂²	1	1	yes	characteristic
26	#14273715	C₂²	1	1	yes	characteristic
27	#14273705	—	2	1	no	—
28	#14273705	—	2	1	no	—
29	#14273705	—	2	1	no	—
30	#14273703	—	2	1	no	—
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	#14273692	C₂²	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
34	#14273689	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	#14273689	C₂²	1	1	yes	characteristic
37	#14273684	—	2	1	no	—
38	#14273660	C₂²	1	1	yes	characteristic
39	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	#14273652	C₂²	1	1	yes	characteristic
42	#14273651	C₂²	1	1	yes	characteristic
43	#14273650	C₂²	1	1	yes	characteristic
44	#14273650	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273647	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273638	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	Q₈×D₅₉	—	4	4	no	—
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	—	2	1	no	—
70	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
71	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₄⋊C₄×C₅₉	—	2	1	no	—
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄²×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	#11745671	—	2	2	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
98	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	D₄₇₂	—	2	2	no	—
105	#11745660	—	2	2	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	C₂²×D₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	—	2	1	no	—
110	Q₈×C₅₉	—	4	2	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	D₂₃₆	—	2	1	no	—
125	D₂₃₆	—	2	1	no	—
126	D₂₃₆	—	2	1	no	—
127	D₂₃₆	#41	1	2	yes	—
128	C₄×D₅₉	—	4	2	no	—
129	C₄×D₅₉	—	4	2	no	—
130	C₄×D₅₉	—	4	2	no	—
131	C₄×D₅₉	—	4	2	no	—
132	Dic₁₁₈	—	2	2	no	—
133	Dic₁₁₈	—	2	1	no	—
134	Dic₁₁₈	—	2	1	no	—
135	Dic₁₁₈	—	4	2	no	—
136	Dic₁₁₈	—	4	2	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₅₉⋊C₈	—	2	1	no	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₅₉	—	2	2	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₂×D₅₉	—	4	1	no	—
145	C₂×D₅₉	—	4	1	no	—
146	C₂×D₅₉	—	4	1	no	—
147	C₂×D₅₉	—	4	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	4	1	no	—
151	C₄×C₅₉	—	4	1	no	—
152	C₄×C₅₉	—	4	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₅₉	—	2	1	no	—
157	Dic₅₉	—	2	1	no	—
158	Dic₅₉	—	2	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
162	#3367	—	59	1	no	maximal, Sylow
163	C₂×C₅₉	#548	1	1	yes	characteristic
164	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₅₉	#578	1	1	yes	characteristic
166	D₅₉	—	4	2	no	—
167	D₅₉	—	8	1	no	—
168	#556	—	59	1	no	—
169	D₄×Q₈	—	59	1	no	—
170	#489	—	59	1	no	—
171	#489	—	59	1	no	—
172	#487	—	59	1	no	—
173	#477	—	59	1	no	—
174	#477	D₅₉	1	1	yes	characteristic, complemented
175	#474	—	59	1	no	—
176	#463	—	59	1	no	—
177	#463	—	59	1	no	—
178	#461	—	59	1	no	—
179	#446	—	59	1	no	—
180	#440	—	59	1	no	—
181	#440	—	59	1	no	—
182	C₄×D₈	—	59	1	no	—
183	C₅₉	#3367	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	118	1	no	—
185	C₂×SD₁₆	—	59	1	no	—
186	C₂×SD₁₆	—	59	1	no	—
187	C₂×D₈	—	59	1	no	—
188	C₄⋊Q₈	—	59	1	no	—
189	C₄⋊Q₈	—	59	1	no	—
190	C₄⋊Q₈	—	59	1	no	—
191	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
192	#126	—	118	1	no	—
193	#126	—	118	1	no	—
194	#124	—	59	1	no	—
195	#124	—	59	1	no	—
196	C₂²⋊Q₈	—	118	1	no	—
197	C₂²⋊Q₈	—	118	1	no	—
198	C₂²⋊Q₈	—	118	1	no	—
199	C₂²⋊Q₈	—	118	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	59	1	no	—
202	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
203	C₄×D₄	—	59	1	no	—
204	C₄×D₄	—	59	1	no	—
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	59	1	no	—
207	#117	—	118	1	no	—
208	#107	—	59	1	no	—
209	#107	C₂×D₅₉	1	1	yes	characteristic
210	#107	C₂×D₅₉	1	1	yes	characteristic
211	C₄⋊C₈	—	59	1	no	—
212	C₄⋊C₈	—	59	1	no	—
213	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	C₂×D₅₉	1	1	yes	characteristic
219	#103	C₂×D₅₉	1	1	yes	characteristic
220	#102	—	59	1	no	—
221	#102	—	59	1	no	—
222	#102	—	59	1	no	—
223	#102	—	59	1	no	—
224	#97	—	59	1	no	—
225	#97	—	59	1	no	—
226	C₄×C₈	—	59	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	59	1	no	—
229	C₂×Q₈	—	59	1	no	—
230	C₂×Q₈	—	118	1	no	—
231	C₂×Q₈	—	118	1	no	—
232	C₂×Q₈	—	236	4	no	—
233	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×D₄	—	59	1	no	—
235	C₂×D₄	—	59	1	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	C₂²×C₄	—	118	1	no	—
239	C₂²×C₄	—	118	1	no	—
240	SD₁₆	—	118	2	no	—
241	SD₁₆	—	118	2	no	—
242	D₈	—	118	2	no	—
243	C₂×C₈	—	59	1	no	—
244	C₂×C₈	—	59	1	no	—
245	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	59	1	no	—
250	C₄⋊C₄	—	59	1	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	59	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	—	118	1	no	—
255	C₄⋊C₄	—	118	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₂²⋊C₄	—	118	1	no	—
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	59	1	no	—
271	C₄²	—	59	1	no	—
272	C₄²	—	118	1	no	—
273	C₄²	C₂²×D₅₉	1	1	yes	characteristic
274	C₂³	—	118	1	no	—
275	C₂³	—	118	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	118	2	no	—
280	Q₈	—	118	1	no	—
281	Q₈	—	118	1	no	—
282	Q₈	—	236	2	no	—
283	Q₈	—	236	2	no	—
284	D₄	—	59	2	no	—
285	D₄	—	118	1	no	—
286	D₄	—	118	1	no	—
287	D₄	—	118	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	59	1	no	—
293	C₂×C₄	—	59	1	no	—
294	C₂×C₄	—	59	1	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	118	1	no	—
297	C₂×C₄	—	118	1	no	—
298	C₂×C₄	—	236	2	no	—
299	C₂×C₄	—	236	2	no	—
300	C₂×C₄	—	236	2	no	—
301	C₂×C₄	—	236	2	no	—
302	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
303	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	118	1	no	—
308	C₈	—	118	1	no	—
309	C₂²	—	118	2	no	—
310	C₂²	—	236	1	no	—
311	C₂²	—	236	1	no	—
312	C₂²	—	236	1	no	—
313	C₂²	—	236	1	no	—
314	C₂²	—	236	1	no	—
315	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	59	2	no	—
323	C₄	—	118	1	no	—
324	C₄	—	118	1	no	—
325	C₄	—	118	1	no	—
326	C₄	—	236	1	no	—
327	C₄	—	236	1	no	—
328	C₄	#41×D₅₉	1	1	yes	characteristic
329	C₄	#14273775	1	1	yes	characteristic
330	C₂	—	472	1	no	—
331	C₂	—	236	2	no	—
332	C₂	#16838062	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
334	C₂	#16838215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#477	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#477	D₅₉
hypocenter	C₅₉	#3367
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16838124	C₂	semidirect
2. #16838067	C₂	semidirect
3. #16837764	C₂	semidirect
4. #16837763	C₂	semidirect
5. #16837751	C₂	semidirect
6. #16837696	C₂	semidirect
7. #16837681	C₂	semidirect
10. #16837606	C₂	semidirect
11. #16837531	C₂	semidirect
12. #16837501	C₂	semidirect
13. #16837500	C₂	semidirect
15. C₅₉×#477	C₂	semidirect
174. #477	D₅₉	semidirect
183. C₅₉	#3367	semidirect